Network creation games have been studied in many different settings recently.
These games are motivated by social networks in which selfish agents want to
construct a connection graph among themselves. Each node wants to minimize
its average or maximum distance to the others, without paying much to
construct the network. Many generalizations have been considered, including
non-uniform interests between nodes, general graphs of allowable edges,
bounded budget agents, etc. In all of these settings, there is no known
constant bound on the price of anarchy. In fact, in many cases, the price of
anarchy can be very large, namely, a constant power of the number of agents.
This means that we have no control on the behavior of network when agents act
selfishly. On the other hand, the price of stability in all these models is
constant, which means that there is chance that agents act selfishly and we
end up with a reasonable social cost.

In this paper, we show how to use an advertising campaign (as introduced in
SODA 2009 [2]) to find such efficient equilibria in
(n, k)-uniform bounded budget connection
game [10]; our result holds for
k = ω(log(n)). More formally, we present
advertising strategies such that, if an α fraction of the agents agree
to cooperate in the campaign, the social cost would be at most
O(1/α) times the optimum cost. This is the first constant bound
on the price of anarchy that interestingly can be adapted to different
settings. We also generalize our method to work in cases that α is not
known in advance. Also, we do not need to assume that the cooperating agents
spend all their budget in the campaign; even a small fraction (β
fraction) of their budget is sufficient to obtain a constant price of anarchy.